Editing Apps/DysonPotential (section)

=====To Second Order=====

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<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2K(\mu) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
E(\mu) 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2\ln \frac{4}{k^'}  + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^5} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
2\biggl[\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma\biggr]  + \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - 1 \biggr]{k'}^2
+ \frac{3^2}{2^5} \biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{7}{6} \biggr] {k'}^4
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{1}{2}\biggr](k')^2
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma\biggr]  + \frac{1}{2^2}\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma - 2 \biggr]
\biggl( \frac{2a}{c}\biggr) \biggl[ 1 
~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~  \cos\chi )\biggr]
+ \frac{3}{2^6} \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{3\ln \Gamma} - 7 \biggr] 4\biggl( \frac{a}{c}\biggr)^2
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2^2}\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{\ln \Gamma} - 1 \biggr]
\biggl( \frac{2a}{c}\biggr) \biggl[ 1 
~-~\cancelto{0}{ \frac{1}{2}\biggl(\frac{a}{c}\biggr)} (2~-~  \cos\chi )\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, ,
</math>
  </td>
</tr>
</table>

where,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln\Gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi )
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, .
</math>
  </td>
</tr>
</table>

Hence,

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  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )   \biggr]
\biggl[ 1 
~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~  \cos\chi )\biggr]
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 
\biggl\{ 
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
\biggr\} 
\biggl\{
1 ~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 
1  +  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi )
\biggr\}
\biggl\{
\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
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<math>~
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 \biggr]
+ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi )  
- \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  
~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2 \biggr]
+  \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi )   
+ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi )  
- \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(\cos\chi - 1) (2~-~ \cos\chi ) 
+\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi +  6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi )  
~-~ \cos\chi 
+ \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr)  - 8 \biggr]
+ \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
24( -1 ~+~ 4\cos\chi -2\cos^2\chi )   
+ 48 (2~-~ \cos\chi )  
- 48\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr)  - 2    \biggr]
(2~-~  \cos\chi )
+ 36\biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr)  - 7 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+  48(\cos\chi - 1) (2~-~ \cos\chi ) 
~+~12 (1-7\cos\chi +  6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
~+~  ( 13- 24\cos\chi   - 2\cos^2\chi )
~-~12\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi )  
~-~ \cos\chi 
+ \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr)  - 8 \biggr]
+ \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ (-24 + 96\cos\chi-48\cos^2\chi)
+ (96~-~ 48\cos\chi )  
~+~ \biggl[2-  \ln\biggl( \frac{2^3c}{a}\biggr)  \biggr]
(96~-~ 48 \cos\chi )
+ \biggl[ 108 \ln\biggl( \frac{2^3c}{a}\biggr)  - 252 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 48(3\cos\chi - \cos^2\chi - 2 )
~+~ (12 - 84\cos\chi +  72\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr)
~+~  ( 13- 24\cos\chi   - 2\cos^2\chi ) ~+~12\cos\chi
~-~12\cos\chi \biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) - \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-24 ~+~ 96\cos\chi -48\cos^2\chi    + 96~-~ 48\cos\chi   
+ 192  
- 252  
-96\cos\chi ~+~  ( 13- 12\cos\chi   - 2\cos^2\chi )
+ (144\cos\chi - 48\cos^2\chi -96 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+36 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]\cos\chi    
~+~(12 - 84\cos\chi +  72\cos^2\chi -96 + 108) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -98\cos^2\chi    
~+~(24 - 48\cos\chi +  72\cos^2\chi ) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl( \frac{2^3c}{a}\biggr) 
+ \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[  \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi 
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -98\cos^2\chi    
~+~24\ln\biggl( \frac{2^3c}{a}\biggr)(1 - 2\cos\chi +  3\cos^2\chi ) 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) 
</math>
  </td>
</tr>
</table>

In an effort to compare this expression with equation (6) from Dyson's (1893b) "Part II", we should make the substitutions,
<div align="center">
<math>~\ln\biggl(\frac{2^3c}{a}\biggr) \rightarrow (\lambda +2)</math> &nbsp;  &nbsp;  &nbsp; and  &nbsp;  &nbsp;  &nbsp; <math>~2\cos^2\chi \rightarrow 1 + \cos2\chi \, .</math>
</div>

This means,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c}\biggr|_{\mathcal{O}(a^2/c^2)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49(1+\cos2\chi )   
~+~24(\lambda + 2)(1 - 2\cos\chi ) 
~+~36(\lambda+2)(1 + \cos2\chi ) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49 -49 \cos2\chi    
~+~24(\lambda + 2 -2\lambda \cos\chi - 4\cos\chi) 
~+~36(\lambda+2 +\lambda\cos 2\chi + 2\cos 2\chi) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
-71 ~+~84\cos\chi -49 -49 \cos2\chi   
~+~24\lambda + 48 -48\lambda \cos\chi - 96\cos\chi 
~+~36\lambda+72 +36\lambda\cos 2\chi + 72\cos 2\chi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{
60\lambda -48\lambda \cos\chi - 12\cos\chi +36\lambda\cos 2\chi + 23\cos 2\chi 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
~+~ \biggl( \frac{a}{c}\biggr)^2\biggl\{
\frac{5\lambda}{16}  - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi  
\biggr\} \, .
</math>
  </td>
</tr>
</table>

This expression differs from the 2<sup>nd</sup>-order term in Dyson's equation (6) by the amount,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta \biggr|_{\mathcal{O}(a^2/c^2)} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{a}{c}\biggr)^2\biggl\{
\frac{5\lambda}{16}  - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi  
\biggr\} 
-
\biggl(\frac{a}{c}\biggr)^2 \biggl\{ \frac{\lambda - \frac{1}{2}}{16} + \frac{3(\lambda + \frac{17}{36})}{16}\cos2\chi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
60\lambda  - (48\lambda + 12)~\cos\chi +(36\lambda+23)\cos 2\chi  
\biggr\} 
-
\frac{1}{16\cdot 12}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 12\lambda - 6 + (36\lambda + 17) \cos2\chi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{
48\lambda + 6  - (48\lambda + 12)~\cos\chi +(6)\cos 2\chi  
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^2\biggl\{
(\cos 2\chi  -1)- (8\lambda + 2)~(1+\cos\chi) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2\biggl\{
(\cos\chi  - 1)- (4\lambda + 1)~ 
\biggr\} (1+\cos\chi)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2 (\cos\chi  - 2- 4\lambda ) (1+\cos\chi)
</math>
  </td>
</tr>
</table>